FOM: history and f.o.m.

[Stephen G Simpson]

I respond to some points in Bill Tait's posting of 31 Mar 98 11:06:12.

Bill Tait writes of:

 > the `unreasonable effectiveness' question): Why is mathematics such
 > a good tool for describing, controlling, predicting, whatever,
 > nature?  Or, Why does reason work?

and invokes evolution to explain it:

 > Before the theory of evolution by natural selection, this surely
 > presented a profound puzzle, that seemed to demand for its solution a
 > grand designer---e.g. in Plato and Leibniz, the view that there is a
 > best order of things and a creator who designed the cosmos with this
 > order and humans with at least a limited propensity to understand it.
 > But surely evolution by natural selection allows us to believe that
 > the explanation for the usefulness of mathematics and for our ability
 > to develop and apply it can be accounted for in the same way as the
 > usefulness of other social institutions and individual physical and
 > mental organs and powers. 

Thanks for this thought-provoking point.  By the way, an excellent
popular book on evolution and the argument from design is "The Blind
Watchmaker" by Richard Dawkins.

It's exciting that Darwinian evolution and natural selection (along
with brain research?) may give us a lead or possible approach to a
precise causal explanation of exactly how our faculty of reason works
and why it is effective.  However, I don't go along with the idea that
pre-Darwinian philosophers and scientists had no option but a Grand
Designer.  It seems to me that this excuse lets these mystics off too
lightly.  What's wrong with simply taking our faculty of reason as a
given?  We know that our minds enable us to understand reality, and
that fact is or should be one of the starting points or axioms of
philosophy (along with the fact that reality is real).  In my view,
it's wrong to base such absolutely fundamental philosophical points on
advanced scientific theories such as evolution; it's an inversion of
the conceptual hierarchy.  To my way of thinking, the premier example
of a pre-Darwinian philosopher who exemplifies the correct approach is
Aristotle.

 > What strikes me about the vertical point of view of reverse
 > mathematics is that it responds to historical accidents---what
 > theorems we have happened to proved. (I don't mean to imply
 > contempt for our human interests; only that, at least as I
 > conceived it, foundations shouldn't take them into account.)  
 ... 
 > I would be very happy (I hope!) to hear the response of Harvey and
 > Steve S. and others to what I have said.

My preliminary response would be that, as Aristotle said, nothing in
mathematics is accidental.  The "theorems we happen to have proved"
are not historical accidents.  I'm deeply convinced that there are
logical reasons why mathematics evolved the way it did.  Reverse
mathematics has helped to elucidate this.

Bill, I sense a kind of inconsistency in your viewpoint.  In
philosophy, you appreciate and extol the value of historical analysis.
But in mathematics, you dismiss "the theorems that we happen to have
proved" as historical accidents, not relevant to foundations.  Why?

-- Steve